MinT - Best Known (139, 139+81, s)-Nets in Base 3

Best Known (139, 139+81, s)-Nets in Base 3

(139, 139+81, 156)-Net over F₃ — Constructive and digital

Digital (139, 220, 156)-net over F₃, using

14 times m-reduction [i] based on digital (139, 234, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 117, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(139, 139+81, 258)-Net over F₃ — Digital

Digital (139, 220, 258)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+81, 3189)-Net in Base 3 — Upper bound on s

There is no (139, 220, 3190)-net in base 3, because

1 times m-reduction [i] would yield (139, 219, 3190)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 311 808831 747153 107117 394631 260888 343515 992174 138709 021013 678977 597232 368803 723449 803363 741779 407072 768017 > 3²¹⁹ [i]